The cigarette is a symbolic tool that is used to show where Micheal's head is at in the films.
In pt.1, he goes to light a cig for Enzio at the hospital and notices how calm his hand under the stress of the moment. In 2, he lights a cigarette in his argument with Kay to check himself to stay in control. In 3, he's trying to step away from that kind of life and in that, disowning that crutch is apart of that. It's symbolic of him really trying to clean up the family.